Find the points on the curve where the tangent is horizontal or vertical.

Question

Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. (Enter your answers as a comma-separated list of ordered pairs.)
x = 10 − t2, y = t3 − 12t

 

Need help with horizontal tangent = 

Answer 1

x = 10 - t^2

x + t^2 = 10

t^2 = 10 - x

t = sqrt(10-x)


y = t^3 - 12t = t ( t^2 - 12)

= sqrt(10 -x) [ 10 - x - 12]

= [ -2 - x] sqrt(10-x)

= -(x+2)sqrt(10-x)

= -(x+2)(10 -x)^1/2


y' = -1/2(x+2)(10-x)^(-1/2) (-1) + (-1) (10-x)^(1/2)

= 1/2(x+2)(10-x)^(-1/2) - (10-x)^(1/2)

= (10-x)^(-1/2) [ 1/2(x+2) - (10-x) ]

= (10-x)^(-1/2) [ 1/2x + 1 - 10 + x]

= (10-x)^(-1/2) [ 3/2x - 9]

= [(3/2)X - 9 ]/ sqrt(10 -x)

Horizontal tangent when (3/2)X - 9 = 0

3/2X = 9

X = 6

Vertical tangent when sqrt(10-x) = 0
10 -x = 0
x = 10